Automatic Generation of Nonlinear 3D-Shell Finite Element Code by Symbolic Manipulations

摘要

The nonlinear analysis of shells has mainly been restricted to "classical" shell model with "Reissner-Mindlin" kinematics. Even in such relatively simplified setting the numerical implementation issues have been finally settled fairly recently; mostly due to complexity of nonlinear kinematics of large rotations. The recent research on the nonlinear shell problem, e.g. [1] and references therein, is attempting to close the gap between the "classical" shell model (which is basically 2D) and the 3D solid model by trying to produce a 3D-shell model which fits in between. Such a 3D-shell model should possibly retain the "classical" shell model computational efficiency and be able to predict the 3D stress field in thicker shells that would be close to the "true" 3D solid stress field. A task needed to be addressed is to identify an optimal 3D-shell model and its optimal finite element implementation among a vast set of possible 3D-shell models and implementation procedures. This can be achieved by developing a model error estimator in a sense of [2], which would indicate an error of a 3D-shell formulation with respect to the 3D solid formulation for a problem at hand. In order to be able to study performance of different 3D-shell finite element formulations, one needs to produce their finite element codes in an efficient way. Symbolic manipulations, more precisely, automatic generator of numerical codes AceGen [3], has been used to that end. The generator is a Mathematica package able to resolve the mayor problem of symbolic generation of nonlinear finite element arrays (e.g. stiffness matrix, etc.), which is an exponential growth of size of the derived expressions. An approach, implemented in AceGen, avoids this problem by combining: symbolic and algebraic capabilities of Mathematica, automatic differentiation technique, simultaneous optimization of expressions and theorem proving. The generator translates final symbolic formulas in a compiled language and incorporates the code into a nonlinear finite element analysis environment (e.g. Feap). The first aim of this paper is to present how the above mentioned automatic generator can produce finite element codes of complex 3-D shell finite rotation formulations efficiently. The second aim is to present the initial results of our work on model error estimation for shell formulations.